Self-orthogonal codes over a non-unital ring and combinatorial matrices

2021/06/14 by Shi, Minjia, Wang, Shukai, Kim, Jon-Lark +1 · 1 citation
#94B05 #F.2.2 #FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.2106.07124

Abstract

There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as E=⟨ a,b | 2a=2b=0, a²=a, b²=b, ab=a, ba=b⟩. We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over E, and Type IV codes, that is, quasi self-dual codes whose all codewords have even Hamming weight. All these codes can be represented as formally self-dual additive codes over \F₄. The classical invariant theory bound for the weight enumerators of this class of codesimproves the known bound on the minimum distance of Type IV codes over E.

Cited by

• A Generic Construction on Self-orthogonal Algebraic Geometric Codes and Its Applications

Related

• Self-Dual Codes over Z₂xZ₄
• Expanding self-orthogonal codes over a ring \Z₄ to self-dual codes and unimodular lattices
• Duality of Codes over Non-unital Rings of Order Six
• Linear Codes over Z₄+uZ₄: MacWilliams identities, projections, and formally self-dual codes
• Self-orthogonal codes from equitable partitions of association schemes